Fun problem: ? x / (x^2 + 6x + 10) dx

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So I would recommend the substitution x= u- 3, dx= du, and the problem becomes \int\frac{u-3}{u^2+1}{\rm d}u = \int\frac{u}{u^2+1}{\rm d}u - \int\frac{3}{u^2+1}{\rm d}uIn summary, the problem of integrating x/(x^2 + 6x + 10) dx can be solved using integration by parts and the substitution u = x + 3. This will result in two integrals, one of which can be solved using the substitution v = u^2 + 1 and the other can be solved using
  • #1
Matt Jacques
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Fun problem: ? x / (x^2 + 6x + 10) dx

Integration by parts proves 1=1! My mathematical fame is at hand! So how would you do this one?
 
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  • #2
[tex]\int\frac{x}{x^2 + 6x + 10}{\rm d}x = \frac12\left(\ln[10 + x(6 + x)]-6 \arctan [3 + x] \right)[/tex]


So, what does this have to do with 1=1 (which is selfevidently true anyway)?
 
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  • #3
Integration by parts proves 1=1? In other words, you used integration by parts twice, the second time reversing your choice for u and dv so the two cancelled!

"Partial fractions" is what you need here. The denominator, x^2 + 6x + 10, is "irreducible" over the real numbers. It is the same as
x^2+ 6x+ 9+ 1= (x+3)^2+ 1. I would recommend the substitution
u= x+ 3 so that du= dx, x= u- 3 and the problem becomes integrating
(u-3)/(u^2+1)= u/(u^2+1)- 3/(u^2+1).

The first of those can be done by the further substitution v= u^2+1 and the second is a simple arctangent.
 
  • #4
OOops, I forgot that x could be expressed in terms of u.
 
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1. What is the purpose of the problem "Fun problem: ? x / (x^2 + 6x + 10) dx"?

The purpose of this problem is to test one's understanding of integration techniques, specifically the use of partial fraction decomposition and substitution, as well as the application of the fundamental theorem of calculus.

2. How do I approach solving this problem?

To solve this problem, you can start by factoring the denominator into two linear factors. Then, use partial fraction decomposition to express the integrand as a sum of simpler fractions. Next, use substitution to simplify the integral and solve for the antiderivative. Finally, don't forget to add the constant of integration.

3. What are some common mistakes to avoid when solving this problem?

Some common mistakes to avoid include forgetting to factor the denominator, making errors in the partial fraction decomposition, forgetting to add the constant of integration, and not double-checking your final answer by differentiating it.

4. Is there a shortcut or easier way to solve this problem?

There is no shortcut or easier way to solve this problem. Integration problems require a thorough understanding of the concepts and techniques involved, and it is important to show all the steps in your solution to receive full credit.

5. How can I check if I have solved this problem correctly?

You can check your solution by differentiating it and verifying that it matches the original integrand. Additionally, you can use online calculators or ask a classmate or instructor to verify your answer.

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